From db030c458e832ea20fb6143c13a2e9b5643da7bf Mon Sep 17 00:00:00 2001
From: DuncDennis <90915296+DuncDennis@users.noreply.github.com>
Date: Thu, 20 Apr 2023 09:49:15 +0200
Subject: [PATCH] WIP: New simulation backend with better structure.

---
 src/lorenzpy/simulations.py                   | 308 ------------------
 src/lorenzpy/simulations/__init__.py          |  62 ++++
 src/lorenzpy/simulations/_autonomous_flows.py |  76 +++++
 src/lorenzpy/simulations/_base.py             | 218 +++++++++++++
 src/lorenzpy/simulations/_discrete_maps.py    |  19 ++
 src/lorenzpy/simulations/_driven_systems.py   |  16 +
 src/lorenzpy/simulations/_others.py           | 234 +++++++++++++
 tests/test_simulations.py                     |   7 +-
 8 files changed, 631 insertions(+), 309 deletions(-)
 delete mode 100644 src/lorenzpy/simulations.py
 create mode 100644 src/lorenzpy/simulations/__init__.py
 create mode 100644 src/lorenzpy/simulations/_autonomous_flows.py
 create mode 100644 src/lorenzpy/simulations/_base.py
 create mode 100644 src/lorenzpy/simulations/_discrete_maps.py
 create mode 100644 src/lorenzpy/simulations/_driven_systems.py
 create mode 100644 src/lorenzpy/simulations/_others.py

diff --git a/src/lorenzpy/simulations.py b/src/lorenzpy/simulations.py
deleted file mode 100644
index 6abcd88..0000000
--- a/src/lorenzpy/simulations.py
+++ /dev/null
@@ -1,308 +0,0 @@
-"""Simulate various continuous and discrete chaotic dynamical system.
-
-Every dynamical system is represented as a class.
-
-The available classes are:
-- Lorenz63
-- MackeyGlass
-
-The system's parameters are introduced in the class's constructor.
-
-For example when creating a system object of the Lorenz63, the Lorenz parameters,
-sigma, rho, beta, and the timestep dt are parsed as:
-
-sys_obj = Lorenz63(sigma=10, rho=10, beta=5, dt=1)
-
-Each sys_obj contains a "simulate" function.
-To simulate 1000 time-steps of the Lorenz63 system call:
-
-sys_obj.simulate(1000).
-
-The general syntax to create a trajectory of a System is given as:
-
-trajectory = <SystemClass>(<parameters>=<default>).
-simulate(time_steps, starting_point=<default>)
-
-Examples:
-    >>> import lorenzpy.simulations as sims
-    >>> data = sims.Lorenz63().simulate(1000)
-    >>> data.shape
-    (1000, 3)
-"""
-
-from __future__ import annotations
-
-from abc import ABC, abstractmethod
-from typing import Callable
-
-import numpy as np
-
-
-def _runge_kutta(
-    f: Callable[[np.ndarray], np.ndarray], dt: float, x: np.ndarray
-) -> np.ndarray:
-    """Simulate one step for ODEs of the form dx/dt = f(x(t)) using Runge-Kutta.
-
-    Args:
-        f: function used to calculate the time derivative at point x.
-        dt: time step size.
-        x: d-dim position at time t.
-
-    Returns:
-       d-dim position at time t+dt.
-
-    """
-    k1: np.ndarray = dt * f(x)
-    k2: np.ndarray = dt * f(x + k1 / 2)
-    k3: np.ndarray = dt * f(x + k2 / 2)
-    k4: np.ndarray = dt * f(x + k3)
-    next_step: np.ndarray = x + 1.0 / 6 * (k1 + 2 * k2 + 2 * k3 + k4)
-    return next_step
-
-
-def _timestep_iterator(
-    f: Callable[[np.ndarray], np.ndarray], time_steps: int, starting_point: np.ndarray
-) -> np.ndarray:
-    """Iterate a function f: x(i+1) = f(x(i)) multiple times to obtain a full traj.
-
-    Args:
-        f: The iterator function x(i+1) = f(x(i)).
-        time_steps: The number of time_steps of the output trajectory.
-                    The starting_point is included as the 0th element in the trajectory.
-        starting_point: Starting point of the trajectory.
-
-    Returns:
-        trajectory: system-state at every simulated timestep.
-
-    """
-    starting_point = np.array(starting_point)
-    traj_size = (time_steps, starting_point.shape[0])
-    traj = np.zeros(traj_size)
-    traj[0, :] = starting_point
-    for t in range(1, traj_size[0]):
-        traj[t] = f(traj[t - 1])
-    return traj
-
-
-class _SimBase(ABC):
-    """A base class for all the simulation classes."""
-
-    default_starting_point: np.ndarray
-    sys_dim: int
-
-    @abstractmethod
-    def iterate(self, x: np.ndarray) -> np.ndarray:
-        """Generate the next time step when the previous one is given."""
-
-    def simulate(
-        self, time_steps: int, starting_point: np.ndarray | None = None
-    ) -> np.ndarray:
-        """Simulate the trajectory.
-
-        Args:
-            time_steps: Number of time steps t to simulate.
-            starting_point: Starting point of the trajectory shape (sys_dim,).
-                            If None, take the default starting point.
-
-        Returns:
-            Trajectory of shape (t, sys_dim).
-
-        """
-        if starting_point is None:
-            starting_point = self.default_starting_point
-        else:
-            if starting_point.size != self.sys_dim:
-                raise ValueError(
-                    "Provided starting_point has the wrong dimension. "
-                    f"{self.sys_dim} was expected and {starting_point.size} "
-                    "was given"
-                )
-        return _timestep_iterator(self.iterate, time_steps, starting_point)
-
-
-class _SimBaseRungeKutta(_SimBase):
-    dt: float
-
-    @abstractmethod
-    def flow(self, x: np.ndarray) -> np.ndarray:
-        """Return the flow of the continuous dynamical system."""
-
-    def iterate(self, x: np.ndarray) -> np.ndarray:
-        """Calculate the next timestep x(t+dt) with given x(t) using runge kutta.
-
-        Args:
-            x: the previous point x(t).
-
-        Returns:
-            : x(t+dt) corresponding to input x(t).
-
-        """
-        return _runge_kutta(self.flow, self.dt, x)
-
-
-class Lorenz63(_SimBaseRungeKutta):
-    """Simulate the 3-dimensional autonomous flow: Lorenz-63 attractor.
-
-    Literature values (Sprott, Julien Clinton, and Julien C. Sprott. Chaos and
-    time-series analysis. Vol. 69. Oxford: Oxford university press, 2003.) for default
-    parameters and starting_point:
-    - Lyapunov exponents: (0.9059, 0.0, -14.5723)
-    - Kaplan-Yorke dimension: 2.06215
-    - Correlation dimension: 2.068 +- 0.086
-    They refer to:
-    - Parameters: {"sigma": 10.0, "rho": 28.0, "beta": 8 / 3}
-    - Starting point: [0.0, -0.01, 9.0]
-    """
-
-    default_parameters = {"sigma": 10.0, "rho": 28.0, "beta": 8 / 3, "dt": 0.05}
-    default_starting_point = np.array([0.0, -0.01, 9.0])
-    sys_dim = 3
-
-    def __init__(
-        self,
-        sigma: float | None = None,
-        rho: float | None = None,
-        beta: float | None = None,
-        dt: float | None = None,
-    ) -> None:
-        """Define the system parameters.
-
-        Args:
-            sigma: 'sigma' parameter in the Lorenz 63 equations.
-            rho: 'rho' parameter in the Lorenz 63 equations.
-            beta: 'beta' parameter in the Lorenz 63 equations.
-            dt: Size of time steps.
-        """
-        self.sigma = self.default_parameters["sigma"] if sigma is None else sigma
-        self.rho = self.default_parameters["rho"] if rho is None else rho
-        self.beta = self.default_parameters["beta"] if beta is None else beta
-        self.dt = self.default_parameters["dt"] if dt is None else dt
-
-    def flow(self, x: np.ndarray) -> np.ndarray:
-        """Calculate (dx/dt, dy/dt, dz/dt) with given (x,y,z) for RK4.
-
-        Args:
-            x: (x,y,z) coordinates. Needs to have shape (3,).
-
-        Returns:
-            : (dx/dt, dy/dt, dz/dt) corresponding to input x.
-
-        """
-        return np.array(
-            [
-                self.sigma * (x[1] - x[0]),
-                x[0] * (self.rho - x[2]) - x[1],
-                x[0] * x[1] - self.beta * x[2],
-            ]
-        )
-
-
-class MackeyGlass:
-    """Simulate the Mackey-Glass delay differential system.
-
-    TODO: Add literature values for Lyapunov etc.
-    TODO: Hint the differences between this class and the other Sim classes (delay).
-    TODO: Check if the structure is really good?
-    TODO: Add Proper Tests.
-    TODO: Decide whether to use the simple forward-euler or RK4-style update.
-
-    Note: As the Mackey-Glass system is a delay-differential equation, the class does
-    not contain a simple iterate function.
-    """
-
-    default_parameters = {"a": 0.2, "b": 0.1, "c": 10, "tau": 23.0, "dt": 0.1}
-    default_starting_point = np.array([0.9])
-    sys_dim = 1
-
-    def __init__(
-        self,
-        a: float | None = None,
-        b: float | None = None,
-        c: int | None = None,
-        tau: float | None = None,
-        dt: float | None = None,
-    ) -> None:
-        """Define the Mackey-Glass system parameters.
-
-        Args:
-            a: "a" parameter of the Mackey-Glass equation.
-            b: "b" parameter of the Mackey-Glass equation.
-            c: "c" parameter of the Mackey-Glass equation.
-            tau: Time delay of Mackey-Glass equation.
-            dt: Size of time steps.
-        """
-        self.a = self.default_parameters["a"] if a is None else a
-        self.b = self.default_parameters["b"] if b is None else b
-        self.c = self.default_parameters["c"] if c is None else c
-        self.tau = self.default_parameters["tau"] if tau is None else tau
-        self.dt = self.default_parameters["dt"] if dt is None else dt
-
-        # The number of time steps between t=0 and t=-tau:
-        self.history_steps = int(self.tau / self.dt)
-
-    def flow_mg(self, x: np.ndarray, x_past: np.ndarray) -> np.ndarray:
-        """Calculate the flow of the Mackey-Glass equation.
-
-        Args:
-            x: The immediate value of the system. Needs to have shape (1,).
-            x_past: The delayed value of the system. Needs to have shape (1,).
-
-        Returns:
-            : The flow corresponding to x and x_past.
-        """
-        return np.array(
-            [self.a * x_past[0] / (1 + x_past[0] ** self.c) - self.b * x[0]]
-        )
-
-    def iterate_mg(self, x: np.ndarray, x_past: np.ndarray) -> np.ndarray:
-        """Calculate the next time step in the Mackey-Glass equation.
-
-        Args:
-            x: The immediate value of the system. Needs to have shape (1,).
-            x_past: The delayed value of the system. Needs to have shape (1,).
-
-        Returns:
-            : The next value given the immediate and delayed values.
-        """
-        return x + self.dt * self.flow_mg(x, x_past)
-        # f = lambda x_use: self.flow_mg(x_use, x_past)
-        # return _runge_kutta(f, self.dt, x)
-
-    def simulate(
-        self, time_steps: int, starting_point: np.ndarray | None = None
-    ) -> np.ndarray:
-        """Simulate the Mackey-Glass trajectory.
-
-        Args:
-            time_steps: Number of time steps t to simulate.
-            starting_point: Starting point of the trajectory shape (sys_dim,).
-                            If None, take the default starting point.
-
-        Returns:
-            Trajectory of shape (t, sys_dim).
-
-        """
-        if starting_point is None:
-            starting_point = self.default_starting_point
-        else:
-            if starting_point.size != self.sys_dim:
-                raise ValueError(
-                    "Provided starting_point has the wrong dimension. "
-                    f"{self.sys_dim} was expected and {starting_point.size} "
-                    "was given"
-                )
-
-        traj_w_hist = np.zeros((self.history_steps + time_steps, self.sys_dim))
-        traj_w_hist[: self.history_steps, :] = np.repeat(
-            starting_point, self.history_steps
-        )[:, np.newaxis]
-        traj_w_hist[self.history_steps, :] = starting_point
-
-        for t in range(1, time_steps):
-            t_shifted = t + self.history_steps
-            traj_w_hist[t_shifted] = self.iterate_mg(
-                traj_w_hist[t_shifted - 1],
-                traj_w_hist[t_shifted - 1 - self.history_steps],
-            )
-
-        return traj_w_hist[self.history_steps :, :]
diff --git a/src/lorenzpy/simulations/__init__.py b/src/lorenzpy/simulations/__init__.py
new file mode 100644
index 0000000..fbc82c5
--- /dev/null
+++ b/src/lorenzpy/simulations/__init__.py
@@ -0,0 +1,62 @@
+"""Simulate various continuous and discrete chaotic dynamical system.
+
+Every dynamical system is represented as a class.
+
+The available classes are:
+- Lorenz63
+- MackeyGlass
+
+The system's parameters are introduced in the class's constructor.
+
+For example when creating a system object of the Lorenz63, the Lorenz parameters,
+sigma, rho, beta, and the timestep dt are parsed as:
+
+sys_obj = Lorenz63(sigma=10, rho=10, beta=5, dt=1)
+
+Each sys_obj contains a "simulate" function.
+To simulate 1000 time-steps of the Lorenz63 system call:
+
+sys_obj.simulate(1000).
+
+The general syntax to create a trajectory of a System is given as:
+
+trajectory = <SystemClass>(<parameters>=<default>).
+simulate(time_steps, starting_point=<default>)
+
+Examples:
+    >>> import lorenzpy.simulations as sims
+    >>> data = sims.Lorenz63().simulate(1000)
+    >>> data.shape
+    (1000, 3)
+
+
+TODO <below>
+    - Probably for each concrete simulation class + public methods. Compare with sklearn
+    - Find out which functionality is missing. E.g. Raising error when wrong values are
+      parsed.
+    - Check where to add proper tests and how to add them efficiently. Fixtures?
+      Parametrization?
+    - Implement all the other dynamical systems.
+    - Check if the names of files and functions make sense?
+    - Add functionality to add your own dynamical system? As my base-classes are
+      protected this is maybe not so easy? -> Make ABC public?
+    - Think about adding NARMA? Maybe I need a random number generator framework.
+    - Check if I can further reduce code duplication. Maybe regarding solvers.
+    - Check for proper doc-generation. It seems that the methods of inhereted members
+      is not implemented yet. See:
+      https://github.com/mkdocstrings/mkdocstrings/issues/78
+"""
+
+from ._autonomous_flows import Lorenz63, Lorenz96
+from ._discrete_maps import Logistic
+from ._driven_systems import SimplestDrivenChaotic
+from ._others import KuramotoSivashinsky, MackeyGlass
+
+__all__ = [
+    "Lorenz63",
+    "Lorenz96",
+    "Logistic",
+    "SimplestDrivenChaotic",
+    "KuramotoSivashinsky",
+    "MackeyGlass",
+]
diff --git a/src/lorenzpy/simulations/_autonomous_flows.py b/src/lorenzpy/simulations/_autonomous_flows.py
new file mode 100644
index 0000000..835e5f2
--- /dev/null
+++ b/src/lorenzpy/simulations/_autonomous_flows.py
@@ -0,0 +1,76 @@
+"""Autonomous flows."""
+import numpy as np
+
+from ._base import _BaseSimFlow
+
+
+class Lorenz63(_BaseSimFlow):
+    """Simulation class for the Lorenz63 system.
+
+    This function is able to simulate the chaotic dynamical system originally
+    introduced by Lorenz.
+
+    Attributes:
+        sigma: Sigma parameter.
+        rho: rho parameter.
+    """
+
+    def __init__(self, sigma=10.0, rho=28.0, beta=8 / 3, dt=0.1, solver="rk4"):
+        """Initialize the Lorenz63.
+
+        Args:
+            sigma: a.
+            rho: b.
+            beta: c.
+            dt: d.
+            solver: e.
+        """
+        super().__init__(dt, solver)
+        self.sigma = sigma
+        self.rho = rho
+        self.beta = beta
+
+    def flow(self, x: np.ndarray) -> np.ndarray:
+        return np.array(
+            [
+                self.sigma * (x[1] - x[0]),
+                x[0] * (self.rho - x[2]) - x[1],
+                x[0] * x[1] - self.beta * x[2],
+            ]
+        )
+
+    def get_default_starting_pnt(self) -> np.ndarray:
+        return np.array([0.0, -0.01, 9.0])
+
+
+class Lorenz96(_BaseSimFlow):
+    """Simulate the n-dimensional Lorenz 96 model."""
+
+    def __init__(
+        self, sys_dim: int = 30, force: float = 8.0, dt: float = 0.05, solver="rk4"
+    ) -> None:
+        super().__init__(dt=dt, solver=solver)
+        self.sys_dim = sys_dim
+        self.force = force
+
+    def get_default_starting_pnt(self) -> np.ndarray:
+        return np.sin(np.arange(self.sys_dim))
+
+    def flow(self, x: np.ndarray) -> np.ndarray:
+        system_dimension = x.shape[0]
+        derivative = np.zeros(system_dimension)
+        # Periodic Boundary Conditions for the 3 edge cases i=1,2,system_dimension
+        derivative[0] = (x[1] - x[system_dimension - 2]) * x[system_dimension - 1] - x[
+            0
+        ]
+        derivative[1] = (x[2] - x[system_dimension - 1]) * x[0] - x[1]
+        derivative[system_dimension - 1] = (x[0] - x[system_dimension - 3]) * x[
+            system_dimension - 2
+        ] - x[system_dimension - 1]
+
+        # TODO: Rewrite using numpy vectorization to make faster
+        for i in range(2, system_dimension - 1):
+            derivative[i] = (x[i + 1] - x[i - 2]) * x[i - 1] - x[i]
+
+        derivative = derivative + self.force
+        return derivative
diff --git a/src/lorenzpy/simulations/_base.py b/src/lorenzpy/simulations/_base.py
new file mode 100644
index 0000000..f3db174
--- /dev/null
+++ b/src/lorenzpy/simulations/_base.py
@@ -0,0 +1,218 @@
+"""Base code for simulations."""
+
+from __future__ import annotations
+
+import inspect
+from abc import ABC, abstractmethod
+from typing import Any, Callable
+
+import numpy as np
+
+
+def _forward_euler(
+    f: Callable[[np.ndarray], np.ndarray], dt: float, x: np.ndarray
+) -> np.ndarray:
+    """Simulate one step for ODEs of the form dx/dt = f(x(t)) using the forward euler.
+
+    Args:
+        f: function used to calculate the time derivative at point x.
+        dt: time step size.
+        x: d-dim position at time t.
+
+    Returns:
+       d-dim position at time t+dt.
+
+    """
+    return x + dt * f(x)
+
+
+def _runge_kutta(
+    f: Callable[[np.ndarray], np.ndarray], dt: float, x: np.ndarray
+) -> np.ndarray:
+    """Simulate one step for ODEs of the form dx/dt = f(x(t)) using Runge-Kutta.
+
+    Args:
+        f: function used to calculate the time derivative at point x.
+        dt: time step size.
+        x: d-dim position at time t.
+
+    Returns:
+       d-dim position at time t+dt.
+
+    """
+    k1: np.ndarray = dt * f(x)
+    k2: np.ndarray = dt * f(x + k1 / 2)
+    k3: np.ndarray = dt * f(x + k2 / 2)
+    k4: np.ndarray = dt * f(x + k3)
+    next_step: np.ndarray = x + 1.0 / 6 * (k1 + 2 * k2 + 2 * k3 + k4)
+    return next_step
+
+
+def _timestep_iterator(
+    f: Callable[[np.ndarray], np.ndarray], time_steps: int, starting_point: np.ndarray
+) -> np.ndarray:
+    """Iterate an iterator-function f: x(i+1) = f(x(i)) multiple times."""
+    starting_point = np.array(starting_point)
+    traj_size = (time_steps, starting_point.shape[0])
+    traj = np.zeros(traj_size)
+    traj[0, :] = starting_point
+    for t in range(1, traj_size[0]):
+        traj[t] = f(traj[t - 1])
+    return traj
+
+
+class _BaseSim(ABC):
+    """Base class for all simulation classes."""
+
+    @classmethod
+    def _get_init_params(cls) -> list[inspect.Parameter]:
+        """Get the init parameter names and default values."""
+        init = cls.__init__
+
+        # introspect the constructor arguments to find the model parameters
+        # to represent
+        init_signature = inspect.signature(init)
+
+        # Consider the constructor parameters excluding 'self'
+        parameters = [p for p in init_signature.parameters.values() if p.name != "self"]
+        return parameters
+
+    @classmethod
+    def get_default_params(cls) -> dict[str, Any]:
+        """Get the default parameters."""
+        parameters = cls._get_init_params()
+        out = dict()
+        for param in parameters:
+            out[param.name] = param.default
+        return out
+
+    def get_params(self) -> dict[str, Any]:
+        """Get the current parameters."""
+        out = dict()
+        parameters = self._get_init_params()
+        for param in parameters:
+            key = param.name
+            value = getattr(self, key)
+            out[key] = value
+        return out
+
+    @abstractmethod
+    def simulate(
+        self,
+        time_steps: int,
+        starting_point: np.ndarray | None = None,
+        transient: int = 0,
+    ) -> np.ndarray:
+        """Simulate the trajectory of the Dynamical System.
+
+        Args:
+            time_steps: Number of time steps in returned trajectory.
+            starting_point: The starting point of the trajectory of shape (sim_dim,).
+                            If None is provided, the default value will be used.
+                            If transient is 0, the first point of the returned
+                            trajectory will be the starting_point.
+            transient: Optional number of transient points to discard before tracking
+                       the returned trajectory.
+
+        Returns:
+            The trajectory of shape (time_steps, sim_dim).
+        """
+
+    @abstractmethod
+    def get_default_starting_pnt(self) -> np.ndarray:
+        """Get the default starting point to be used in the simulate function."""
+
+
+class _BaseSimIterate(_BaseSim):
+    """Base class for Simulation classes using an iterate function."""
+
+    @abstractmethod
+    def iterate(self, x: np.ndarray) -> np.ndarray:
+        """Iterate the system from one time step to the next."""
+
+    def simulate(
+        self,
+        time_steps: int,
+        starting_point: np.ndarray | None = None,
+        transient: int = 0,
+    ) -> np.ndarray:
+        """Simulate the trajectory.
+
+        Args:
+            time_steps: Number of time steps t to simulate.
+            starting_point: Starting point of the trajectory shape (sys_dim,).
+                            If None, take the default starting point.
+            transient: TBD
+
+        Returns:
+            Trajectory of shape (t, sys_dim).
+
+        """
+        if starting_point is None:
+            starting_point = self.get_default_starting_pnt()
+
+        return _timestep_iterator(self.iterate, time_steps + transient, starting_point)[
+            transient:, :
+        ]
+
+
+class _BaseSimFlow(_BaseSimIterate):
+    """Base class for continuous-time Simulation classes defined by a flow."""
+
+    @abstractmethod
+    def __init__(
+        self,
+        dt: float,
+        solver: str | Callable[[Callable, float, np.ndarray], np.ndarray],
+    ):
+        """Initialize the time step dt and the solver."""
+        self.dt = dt
+        self.solver = solver
+
+    @abstractmethod
+    def flow(self, x: np.ndarray) -> np.ndarray:
+        """Return the flow of the continuous dynamical system."""
+
+    def iterate(self, x: np.ndarray) -> np.ndarray:
+        """Calculate the next timestep x(t+dt) with given x(t) using runge kutta.
+
+        Args:
+            x: the previous point x(t).
+
+        Returns:
+            : x(t+dt) corresponding to input x(t).
+
+        """
+        if isinstance(self.solver, str):
+            if self.solver == "rk4":
+                x_next = _runge_kutta(self.flow, self.dt, x)
+            elif self.solver == "forward_euler":
+                x_next = _forward_euler(self.flow, self.dt, x)
+            else:
+                raise ValueError(f"Unsupported solver: {self.solver}")
+        else:
+            x_next = self.solver(self.flow, self.dt, x)
+
+        return x_next
+
+
+class _BaseSimFlowDriven(_BaseSimFlow, ABC):
+    """Base class for driven continuous time dynamical systems."""
+
+    def simulate(
+        self,
+        time_steps: int,
+        starting_point: np.ndarray | None = None,
+        transient: int = 0,
+        time_included: bool = False,
+    ) -> np.ndarray:
+        if starting_point is not None:
+            if not time_included:
+                starting_point = np.append(starting_point, 0)
+        traj = super().simulate(
+            time_steps, starting_point=starting_point, transient=transient
+        )
+        if not time_included:
+            return traj[:, :-1]
+        else:
+            return traj
diff --git a/src/lorenzpy/simulations/_discrete_maps.py b/src/lorenzpy/simulations/_discrete_maps.py
new file mode 100644
index 0000000..01713b2
--- /dev/null
+++ b/src/lorenzpy/simulations/_discrete_maps.py
@@ -0,0 +1,19 @@
+"""Discrete maps."""
+import numpy as np
+
+from ._base import _BaseSimIterate
+
+
+class Logistic(_BaseSimIterate):
+    def __init__(self, r: float = 4.0) -> None:
+        self.r = r
+
+    def iterate(self, x: np.ndarray) -> np.ndarray:
+        return np.array(
+            [
+                self.r * x[0] * (1 - x[0]),
+            ]
+        )
+
+    def get_default_starting_pnt(self) -> np.ndarray:
+        return np.array([0.1])
diff --git a/src/lorenzpy/simulations/_driven_systems.py b/src/lorenzpy/simulations/_driven_systems.py
new file mode 100644
index 0000000..f19dce5
--- /dev/null
+++ b/src/lorenzpy/simulations/_driven_systems.py
@@ -0,0 +1,16 @@
+"""Driven Systems."""
+import numpy as np
+
+from ._base import _BaseSimFlowDriven
+
+
+class SimplestDrivenChaotic(_BaseSimFlowDriven):
+    def __init__(self, omega: float = 1.88, dt: float = 0.1, solver="rk4") -> None:
+        super().__init__(dt, solver)
+        self.omega = omega
+
+    def flow(self, x: np.ndarray) -> np.ndarray:
+        return np.array([x[1], -(x[0] ** 3) + np.sin(self.omega * x[2]), 1])
+
+    def get_default_starting_pnt(self) -> np.ndarray:
+        return np.array([0.0, 0.0, 0.0])
diff --git a/src/lorenzpy/simulations/_others.py b/src/lorenzpy/simulations/_others.py
new file mode 100644
index 0000000..41e90f3
--- /dev/null
+++ b/src/lorenzpy/simulations/_others.py
@@ -0,0 +1,234 @@
+"""Dynamical systems implemented with other algorithms."""
+from __future__ import annotations
+
+from typing import Callable
+
+import numpy as np
+
+from ._base import _BaseSim, _BaseSimIterate, _forward_euler, _runge_kutta
+
+
+class KuramotoSivashinsky(_BaseSimIterate):
+    """Simulate the n-dimensional Kuramoto-Sivashinsky PDE.
+
+    Note: dimension must be an even number.
+
+    PDE: y_t = -y*y_x - (1+eps)*y_xx - y_xxxx.
+
+    Reference for the numerical integration:
+    "fourth order time stepping for stiff pde-kassam trefethen 2005" at
+    https://people.maths.ox.ac.uk/trefethen/publication/PDF/2005_111.pdf
+
+    Python implementation at: https://github.com/E-Renshaw/kuramoto-sivashinsky
+
+    Literature values (doi:10.1017/S1446181119000105) for Lyapunov Exponents:
+    - lyapunov exponents: (0.080, 0.056, 0.014, 0.003, -0.003 ...)
+    They refer to:
+    - Parameters: {"sys_length": 36.0, "eps": 0.0}
+    """
+
+    def __init__(
+        self,
+        sys_dim: int = 50,
+        sys_length: float = 36.0,
+        eps: float = 0.0,
+        dt: float = 0.1,
+    ) -> None:
+        self.sys_dim = sys_dim
+        self.sys_length = sys_length
+        self.eps = eps
+        self.dt = dt
+
+        self._prepare()
+
+    def get_default_starting_pnt(self) -> np.ndarray:
+        x = (
+            self.sys_length
+            * np.transpose(np.conj(np.arange(1, self.sys_dim + 1)))
+            / self.sys_dim
+        )
+        return np.array(
+            np.cos(2 * np.pi * x / self.sys_length)
+            * (1 + np.sin(2 * np.pi * x / self.sys_length))
+        )
+
+    def _prepare(self) -> None:
+        """Calculate internal attributes.
+
+        TODO: make auxiliary variables protected.
+        """
+        k = (
+            np.transpose(
+                np.conj(
+                    np.concatenate(
+                        (
+                            np.arange(0, self.sys_dim / 2),
+                            np.array([0]),
+                            np.arange(-self.sys_dim / 2 + 1, 0),
+                        )
+                    )
+                )
+            )
+            * 2
+            * np.pi
+            / self.sys_length
+        )
+
+        L = (1 + self.eps) * k**2 - k**4
+
+        self.E = np.exp(self.dt * L)
+        self.E_2 = np.exp(self.dt * L / 2)
+        M = 64
+        r = np.exp(1j * np.pi * (np.arange(1, M + 1) - 0.5) / M)
+        LR = self.dt * np.transpose(np.repeat([L], M, axis=0)) + np.repeat(
+            [r], self.sys_dim, axis=0
+        )
+        self.Q = self.dt * np.real(np.mean((np.exp(LR / 2) - 1) / LR, axis=1))
+        self.f1 = self.dt * np.real(
+            np.mean((-4 - LR + np.exp(LR) * (4 - 3 * LR + LR**2)) / LR**3, axis=1)
+        )
+        self.f2 = self.dt * np.real(
+            np.mean((2 + LR + np.exp(LR) * (-2 + LR)) / LR**3, axis=1)
+        )
+        self.f3 = self.dt * np.real(
+            np.mean((-4 - 3 * LR - LR**2 + np.exp(LR) * (4 - LR)) / LR**3, axis=1)
+        )
+
+        self.g = -0.5j * k
+
+    def iterate(self, x: np.ndarray) -> np.ndarray:
+        """Calculate next timestep x(t+1) with given x(t).
+
+        Args:
+            x: (x_0(i),x_1(i),..) coordinates. Needs to have shape (self.sys_dim,).
+
+        Returns:
+            : (x_0(i+1),x_1(i+1),..) corresponding to input x.
+
+        """
+        v = np.fft.fft(x)
+        Nv = self.g * np.fft.fft(np.real(np.fft.ifft(v)) ** 2)
+        a = self.E_2 * v + self.Q * Nv
+        Na = self.g * np.fft.fft(np.real(np.fft.ifft(a)) ** 2)
+        b = self.E_2 * v + self.Q * Na
+        Nb = self.g * np.fft.fft(np.real(np.fft.ifft(b)) ** 2)
+        c = self.E_2 * a + self.Q * (2 * Nb - Nv)
+        Nc = self.g * np.fft.fft(np.real(np.fft.ifft(c)) ** 2)
+        v = self.E * v + Nv * self.f1 + 2 * (Na + Nb) * self.f2 + Nc * self.f3
+        return np.real(np.fft.ifft(v))
+
+
+class MackeyGlass(_BaseSim):
+    """Simulate the Mackey-Glass delay differential system.
+
+    TODO: Add literature values for Lyapunov etc.
+    TODO: Hint the differences between this class and the other Sim classes (delay).
+    TODO: Check if the structure is really good?
+    TODO: Add Proper Tests.
+    TODO: Decide whether to use the simple forward-euler or RK4-style update.
+
+    Note: As the Mackey-Glass system is a delay-differential equation, the class does
+    not contain a simple iterate function.
+    """
+
+    def __init__(
+        self,
+        a: float = 0.2,
+        b: float = 0.1,
+        c: int = 10,
+        tau: float = 23.0,
+        dt: float = 0.1,
+        solver: str | Callable = "rk4",
+    ) -> None:
+        self.a = a
+        self.b = b
+        self.c = c
+        self.tau = tau
+        self.dt = dt
+        self.solver = solver
+
+        # The number of time steps between t=0 and t=-tau:
+        self.history_steps = int(self.tau / self.dt)
+
+    def get_default_starting_pnt(self) -> np.ndarray:
+        return np.array([0.9])
+
+    def flow_mg(self, x: np.ndarray, x_past: np.ndarray) -> np.ndarray:
+        """Calculate the flow of the Mackey-Glass equation.
+
+        Args:
+            x: The immediate value of the system. Needs to have shape (1,).
+            x_past: The delayed value of the system. Needs to have shape (1,).
+
+        Returns:
+            : The flow corresponding to x and x_past.
+        """
+        return np.array(
+            [self.a * x_past[0] / (1 + x_past[0] ** self.c) - self.b * x[0]]
+        )
+
+    def iterate_mg(self, x: np.ndarray, x_past: np.ndarray) -> np.ndarray:
+        """Calculate the next time step in the Mackey-Glass equation.
+
+        Args:
+            x: The immediate value of the system. Needs to have shape (1,).
+            x_past: The delayed value of the system. Needs to have shape (1,).
+
+        Returns:
+            : The next value given the immediate and delayed values.
+        """
+
+        def flow_like(x_use: np.ndarray) -> np.ndarray:
+            return self.flow_mg(x_use, x_past)
+
+        if isinstance(self.solver, str):
+            if self.solver == "rk4":
+                x_next = _runge_kutta(flow_like, self.dt, x)
+            elif self.solver == "forward_euler":
+                x_next = _forward_euler(flow_like, self.dt, x)
+            else:
+                raise ValueError(f"Unsupported solver: {self.solver}")
+        else:
+            x_next = self.solver(flow_like, self.dt, x)
+
+        return x_next
+
+    def simulate(
+        self,
+        time_steps: int,
+        starting_point: np.ndarray | None = None,
+        transient: int = 0,
+    ) -> np.ndarray:
+        """Simulate the Mackey-Glass trajectory.
+
+        Args:
+            time_steps: Number of time steps t to simulate.
+            starting_point: Starting point of the trajectory shape (sys_dim,).
+                            If None, take the default starting point.
+            transient: Washout before storing the trajectory.
+
+        Returns:
+            Trajectory of shape (t, sys_dim).
+        """
+        if starting_point is None:
+            starting_point = self.get_default_starting_pnt()
+
+        if starting_point.size == self.history_steps + 1:
+            initial_history = starting_point
+        elif starting_point.size == 1:
+            initial_history = np.repeat(starting_point, self.history_steps)
+        else:
+            raise ValueError("Wrong size of starting point.")
+
+        traj_w_hist = np.zeros((self.history_steps + time_steps, 1))
+        traj_w_hist[: self.history_steps, :] = initial_history[:, np.newaxis]
+        traj_w_hist[self.history_steps, :] = starting_point
+
+        for t in range(1, time_steps + transient):
+            t_shifted = t + self.history_steps
+            traj_w_hist[t_shifted] = self.iterate_mg(
+                traj_w_hist[t_shifted - 1],
+                traj_w_hist[t_shifted - 1 - self.history_steps],
+            )
+
+        return traj_w_hist[self.history_steps + transient :, :]
diff --git a/tests/test_simulations.py b/tests/test_simulations.py
index 78bffe0..40602e9 100644
--- a/tests/test_simulations.py
+++ b/tests/test_simulations.py
@@ -3,9 +3,10 @@
 import pytest
 
 from lorenzpy import simulations
+from lorenzpy.simulations._base import _BaseSimIterate
 
 
-class DemoSim(simulations._SimBase):
+class DemoSim(_BaseSimIterate):
     """A simple simulation class subclassing simulations._SimBase."""
 
     sys_dim = 3
@@ -18,6 +19,10 @@ def iterate(self, x: np.ndarray) -> np.ndarray:
         """Just return the same value again."""
         return x
 
+    def get_default_starting_pnt(self) -> np.ndarray:
+        """Return some default starting point."""
+        return np.array([0, 0, 0])
+
 
 def test_lorenz63_simulation_shape():
     """Testing lorenz63 simulation."""